GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions

Yang Li, Xu Wu, Jiatu Yan, Jiang Yang, Qinghai Zhang, Shubo Zhao
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Abstract

Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007 Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a) electric boundary conditions with no explicit enforcement of the no-penetration condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the divergence of an initially non-solenoidal velocity, and (d) monotonic decrease of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES formulations are of strong forms and are designed for finite volume/difference methods under the framework of method of lines. Furthermore, we develop semi-discrete algorithms that preserve (c) and (d) and fully discrete algorithms that are fourth-order accurate for velocity both in time and in space. These algorithms employ algebraically stable time integrators in a black-box manner and only consist of solving a sequence of linear equations in each time step. Results of numerical tests confirm our analysis.
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GePUP-ES:具有无滑动条件的不可压缩纳维-斯托克斯方程的高阶能量稳定投影方法
受无约束 PPE (UPPE) 公式 [Liu, Liu, & Pego 2007Comm. Pure Appl. Math., 60 pp. 1443] 的启发,我们之前提出了 GePUP 公式 [Zhang 2016 J. Sci.在本文中,我们提出了 GePUP-E 和 GePUP-ES,它们是 GePUP 的变体,具有以下特点:(a)不明确执行无渗透条件的电边界条件;(b)等效于无滑动 INSE;(c)初始非滑动速度的发散呈指数衰减;(d)动能单调递减。与 UPPE 不同的是,GePUP-E 和 GePUP-ES 形式是强形式的,是在线性方法框架下为有限体积/差分方法设计的。此外,我们还开发了保留(c)和(d)的半离散算法和完全离散算法,这些算法在时间和空间上对速度都有四阶精度。这些算法以黑箱方式使用代数稳定的时间积分器,只需求解每个时间步的线性方程序列。数值测试结果证实了我们的分析。
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